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Theoretical and Practical Introduction to COMSOL Multiphysics

Brief Selective Summary of the Short Course

TOMASZ G. ZIELI ´

NSKI bluebox.ippt.pan.pl/˜tzielins/

Institute of Fundamental Technological Research

• f the Polish Academy of Sciences

Warsaw • Poland

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Website, Lecture Notes, Contact

Introductory Course on Multiphysics Modelling

http://bluebox.ippt.pan.pl/˜tzielins/index.php?im=1&id=lectures.html

Go to: http://bluebox.ippt.pan.pl/˜tzielins/ Then, choose: Lectures. Suggested Lecture Notes: · · ·

6 Introduction to Finite Element Method 7 Heat Transfer Problems 8 Galerkin Finite Element Model for Heat Transfer

· · ·

15 Elementary Viscous Flow

· · · Contact:

TOMASZ G. ZIELI ´

NSKI, DSc, PhD, MSc

Institute of Fundamental Technological Research of the Polish Academy of Sciences website at IPPT PAN: http://www.ippt.pan.pl/en/staff/tzielins e-mail: tzielins@ippt.pan.pl

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A typical mathematical modelling process

1

Define the problem geometry – specify: space dimension ([0D, discrete,] 1D, 2D, 3D, mixed) domain or subdomains, boundaries and interfaces between subdomains.

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A typical mathematical modelling process

1

Define the problem geometry – specify:

space dimension ([0D, discrete,] 1D, 2D, 3D, mixed) domain or subdomains, boundaries and interfaces between subdomains.

2

Choose/derive a mathematical model decide on transient (time-dependent) or steady-state analysis; choose problem variables/fields (primary and secondary ones, eg.: concentration and flux, or temperature and heat flux vector); use or derive model equations (usually in terms of Partial Differential Equations, e.g., the diffusion equation); specify material(s) properties, define sources (e.g., heat sources

• r sinks) or excitations (e.g., external forces);

specify boundary conditions (and initial conditions); define couplings on interfaces between different subdomains.

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A typical mathematical modelling process

1

Define the problem geometry – specify:

space dimension ([0D, discrete,] 1D, 2D, 3D, mixed) domain or subdomains, boundaries and interfaces between subdomains.

2

Choose/derive a mathematical model

decide on transient (time-dependent) or steady-state analysis; choose problem variables/fields (primary and secondary ones, eg.: concentration and flux, or temperature and heat flux vector); use or derive model equations (usually in terms of Partial Differential Equations, e.g., the diffusion equation); specify material(s) properties, define sources (e.g., heat sources or sinks)

• r excitations (e.g., external forces);

specify boundary conditions (and initial conditions); define couplings on interfaces between different subdomains.

3

Implement the model and solve the problem choose a method (analytical if possible, or a numerical one); set: the geometry (and time range), material parameters, sources, boundary (and initial) conditions; specify features of the method (e.g., approximation functions, mesh, [time step,] etc.) and solve.

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A typical mathematical modelling process

1

Define the problem geometry – specify:

space dimension ([0D, discrete,] 1D, 2D, 3D, mixed) domain or subdomains, boundaries and interfaces between subdomains.

2

Choose/derive a mathematical model

decide on transient (time-dependent) or steady-state analysis; choose problem variables/fields (primary and secondary ones, eg.: concentration and flux, or temperature and heat flux vector); use or derive model equations (usually in terms of Partial Differential Equations, e.g., the diffusion equation); specify material(s) properties, define sources (e.g., heat sources or sinks)

• r excitations (e.g., external forces);

specify boundary conditions (and initial conditions); define couplings on interfaces between different subdomains.

3

Implement the model and solve the problem

choose a method (analytical if possible, or a numerical one); set: the geometry (and time range), material parameters, sources, boundary (and initial) conditions; specify features of the method (e.g., approximation functions, mesh, [time step,] etc.) and solve.

4

Post-process the results of solution and draw conclusions from the model predictions (re-design, optimise, etc.).

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Basics of multi-scale modelling

Motivation: Many complex phenomena involve processes occurring at different scales (of space and/or time), or . . . . . . multiple spatial and/or temporal scales can be distinguished to differ between the process phases or to better/easier describe the process features. Usually, it is easier to deal with different scales individually.

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Basics of multi-scale modelling

Motivation: Many complex phenomena involve processes occurring at different scales (of space and/or time), or . . . . . . multiple spatial and/or temporal scales can be distinguished to differ between the process phases or to better/easier describe the process features. Usually, it is easier to deal with different scales individually. Multi-scale modelling Mathematical solution techniques of dealing with problems that have important features at multiple scales of space and/or time. Comment: For many problems, the processes (i.e., sub-problems) at various scales can be, in practice, solved (quasi) separately, which makes such multi-scale approach very efficient.

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Basics of multi-scale modelling

Multi-scale modelling Mathematical solution techniques of dealing with problems that have important features at multiple scales of space and/or time. Requirements: Separation of scales – allows to apply different approaches to treat problems at various scales. One can distinguish:

different spatial scales – when there are local and global phenomena,

• r there co-exist processes which are: essentially microscopic (i.e.,
• ccur at the micro-scale), mesoscopic (i.e., occur at the meso-scale),

and macroscopic (i.e., occur at the macro-scale), etc.; different temporal scales – when the involved processes are: relatively slow (static or quasi-static), dynamic, or relatively fast, etc.

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Basics of multi-scale modelling

Multi-scale modelling Mathematical solution techniques of dealing with problems that have important features at multiple scales of space and/or time. Requirements: Separation of scales – allows to apply different approaches to treat problems at various scales. One can distinguish:

different spatial scales – when there are local and global phenomena,

• r there co-exist processes which are: essentially microscopic (i.e.,
• ccur at the micro-scale), mesoscopic (i.e., occur at the meso-scale),

and macroscopic (i.e., occur at the macro-scale), etc.; different temporal scales – when the involved processes are: relatively slow (static or quasi-static), dynamic, or relatively fast, etc.

Representativeness of the geometry or time-interval for the phenomenon considered on the scale related to this geometry or time-interval.

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Basics of multi-scale modelling

Multi-scale modelling Mathematical solution techniques of dealing with problems that have important features at multiple scales of space and/or time. Requirements: Separation of scales – allows to apply different approaches to treat problems at various scales. One can distinguish:

different spatial scales – when there are local and global phenomena,

• r there co-exist processes which are: essentially microscopic (i.e.,
• ccur at the micro-scale), mesoscopic (i.e., occur at the meso-scale),

and macroscopic (i.e., occur at the macro-scale), etc.; different temporal scales – when the involved processes are: relatively slow (static or quasi-static), dynamic, or relatively fast, etc.

Representativeness of the geometry or time-interval for the phenomenon considered on the scale related to this geometry or time-interval. Well defined way of passing of the relevant information (effective properties, behaviour, etc.) between the scales.

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Basics of multi-scale modelling

EXAMPLE: Transport through a porous medium MACRO-SCALE viscous flow through a porous material fluid fluid porous material material with complex microstructure of open pore network saturated with fluid

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Basics of multi-scale modelling

EXAMPLE: Transport through a porous medium MACRO-SCALE MICRO-SCALE (skeleton) Periodic REV fluid fluid porous material R E V

1 MICRO-SCALE:

Selection (construction) of a (periodic) Representative Elementary Volume (REV) of a porous medium.

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Basics of multi-scale modelling

EXAMPLE: Transport through a porous medium MACRO-SCALE MICRO-SCALE (fluid domain) Periodic REV fluid fluid porous material R E V

1 MICRO-SCALE:

Selection (construction) of a (periodic) Representative Elementary Volume (REV) of a porous medium.

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Basics of multi-scale modelling

EXAMPLE: Transport through a porous medium MACRO-SCALE MICRO-SCALE (Stokes flow) Periodic REV fluid fluid porous material R E V

1 MICRO-SCALE:

Selection (construction) of a (periodic) Representative Elementary Volume (REV) of a porous medium. Stokes flow, i.e., linear & steady, viscous, incompressible flow through the periodic RVE, driven by a uniform pressure gradient.

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Basics of multi-scale modelling

EXAMPLE: Transport through a porous medium MACRO-SCALE MICRO-SCALE (Stokes flow) Periodic REV fluid fluid porous material R E V Permeability

1 MICRO-SCALE:

Selection (construction) of a (periodic) Representative Elementary Volume (REV) of a porous medium. Stokes flow, i.e., linear & steady, viscous, incompressible flow through the periodic RVE, driven by a uniform pressure gradient. Averaging of the computed velocity field to determine the permeability of the porous medium.

2 MACRO-SCALE:

Macroscopic flow through the porous material characterised by its open porosity and permeability using the Darcy’s law.

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Questions & Answers

1

What is steady-state (stationary) problem? A system is in steady state if its recently observed behaviour will continue into the future, so that time can be eliminated from the problem description, which means that the corresponding stationary problem is time-independent (i.e. the problem variables do not depend on time). Examples: static problems, steady-state flow, time-harmonic problems.

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Questions & Answers

1

What is steady-state (stationary) problem?

2

What is transient (non-stationary) problem? A system is in transient state where it substantially changes over time, which means that the problem is essentially time-dependent (i.e. the problem variables depend on time). The transient state is often a start-up in many steady state systems.

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Questions & Answers

1

What is steady-state (stationary) problem?

2

What is transient (non-stationary) problem?

3

What is Boundary Value Problem? A Boundary Value Problem (BVP) is a Partial Differential Equation (PDE) – defined on a specified domain – together with appropriate boundary conditions – defined on the domain boundary. BVPs are (time-independent) mathematical models of most steady-state physical phenomena.

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Questions & Answers

1

What is steady-state (stationary) problem?

2

What is transient (non-stationary) problem?

3

What is Boundary Value Problem?

4

What is Initial Boundary Value Problem? An Initial Boundary Value Problem (IBVP) is defined by a time-dependent Partial Differential Equation (PDE) with appropriate initial and boundary conditions. IBVPs are (time-dependent) mathematical models of most transient physical phenomena.

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Questions & Answers

1

What is steady-state (stationary) problem?

2

What is transient (non-stationary) problem?

3

What is Boundary Value Problem?

4

What is Initial Boundary Value Problem?

5

What are mechanisms of heat transfer? Three mechanisms of heat transfer:

I

the conduction – the heat transfer by diffusion,

II

the convection (advection) – the heat transfer due to the bulk movement of fluid,

III the radiation – the heat transfer via electromagnetic waves.

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Questions & Answers

1

What is steady-state (stationary) problem?

2

What is transient (non-stationary) problem?

3

What is Boundary Value Problem?

4

What is Initial Boundary Value Problem?

5

What are mechanisms of heat transfer?

6

What is a usual modelling procedure using Finite Element Method? Major steps in modelling using Finite Element Method:

I

Define the problem geometry: decide on 2D or 3D; if possible, take advantage of symmetry; construct domain (or subdomains) with well-defined boundaries (and interfaces).

II

Choose a mathematical model: decide on steady state (BVP) or transient state (IBVP); specify material(s), sources (or excitations), boundary conditions (and initial conditions in the case of IVBP).

III Construct (or generate) a finite element mesh. IV Solve the problem numerically (choose a numerical solver; in the

case of IBVP set the time scope and time step).

V Post-process and interpret the results (draw conclusions, etc.).

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Questions & Answers

1

What is steady-state (stationary) problem?

2

What is transient (non-stationary) problem?

3

What is Boundary Value Problem?

4

What is Initial Boundary Value Problem?

5

What are mechanisms of heat transfer?

6

What is a usual modelling procedure using Finite Element Method?

7

When can multi-scale modelling be applied (i.e. to what problems)? Multi-scale modelling can be applied to problems that have important features at multiple scales of space and/or time.

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Questions & Answers

1

What is steady-state (stationary) problem?

2

What is transient (non-stationary) problem?

3

What is Boundary Value Problem?

4

What is Initial Boundary Value Problem?

5

What are mechanisms of heat transfer?

6

What is a usual modelling procedure using Finite Element Method?

7

When can multi-scale modelling be applied (i.e. to what problems)?

8

What are the two main requirements in multi-scale modelling? Two main requirements in multi-scale modelling are:

I

separation of scales,

II

representativeness of the geometry domain (or time-interval) of a specific scale for the phenomenon considered on that scale.

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Questions & Answers

1

What is steady-state (stationary) problem?

2

What is transient (non-stationary) problem?

3

What is Boundary Value Problem?

4

What is Initial Boundary Value Problem?

5

What are mechanisms of heat transfer?

6

What is a usual modelling procedure using Finite Element Method?

7

When can multi-scale modelling be applied (i.e. to what problems)?

8

What are the two main requirements in multi-scale modelling?

9

What are two types of scales suitable for multi-scale modelling? Multi-scale modeling can be performed with respect to:

I

spatial scales and/or

II

temporal scales.

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Questions & Answers

1

What is steady-state (stationary) problem?

2

What is transient (non-stationary) problem?

3

What is Boundary Value Problem?

4

What is Initial Boundary Value Problem?

5

What are mechanisms of heat transfer?

6

What is a usual modelling procedure using Finite Element Method?

7

When can multi-scale modelling be applied (i.e. to what problems)?

8

What are the two main requirements in multi-scale modelling?

9

What are two types of scales suitable for multi-scale modelling?

10 Which problems are more difficult (i.e. more computationally

demanding) to solve: compressible or incompressible flows? Compressible flows are more complex (and coupled with thermal problem) and therefore computationally more demanding than incompressible flows.

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Questions & Answers

1

What is steady-state (stationary) problem?

2

What is transient (non-stationary) problem?

3

What is Boundary Value Problem?

4

What is Initial Boundary Value Problem?

5

What are mechanisms of heat transfer?

6

What is a usual modelling procedure using Finite Element Method?

7

When can multi-scale modelling be applied (i.e. to what problems)?

8

What are the two main requirements in multi-scale modelling?

9

What are two types of scales suitable for multi-scale modelling?

10 Which problems are more difficult (i.e. more computationally

demanding) to solve: compressible or incompressible flows?

11 What is the Stokes flow?

Stokes flow is a linearised, steady-state, viscous and incompressible flow (of Newtonian fluid).

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Questions & Answers

1

What is steady-state (stationary) problem?

2

What is transient (non-stationary) problem?

3

What is Boundary Value Problem?

4

What is Initial Boundary Value Problem?

5

What are mechanisms of heat transfer?

6

What is a usual modelling procedure using Finite Element Method?

7

When can multi-scale modelling be applied (i.e. to what problems)?

8

What are the two main requirements in multi-scale modelling?

9

What are two types of scales suitable for multi-scale modelling?

10 Which problems are more difficult (i.e. more computationally

demanding) to solve: compressible or incompressible flows?

11 What is the Stokes flow? 12 What is the fluid-structure interaction (FSI)?

Fluid–structure interaction is the interaction of some movable and/or deformable structure with an internal or surrounding fluid flow.